5333-Solns-Assignment2-Sp13 - MATH 5333 Assignment#2 Section 12 3 Let s denote sup S and let m denote max S(a s = m = 3(c s = m = 4(e s = m = 1(g s = 1

5333-Solns-Assignment2-Sp13 - MATH 5333 Assignment#2...

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MATH 5333: Assignment #2 Section 12 3 Let s denote sup S , and let m denote max S . (a) s = m = 3. (b) s = m = π . (c) s = m = 4. (d) s = 4, no max. (e) s = m = 1. (f) s = 1, no max. (g) s = 1, no max. (h) s = m = 3 / 2. (i) None. (j) s = 4, no max. (k) s = m = 1. (l) s = 2, no max. (m) s = 5, no max (n) s = 5, no max. 4 Let t denote inf S , and let k denote min S . (a) t = k = 1. (b) t = k = 3. (c) t = k = 0. (d) t = 0, no min. (e) t = 0, no min. (f) t = k = 0. (g) t = k = 1 / 2. (h) t = k = - 2. (i) t = k = 0. (j) Neither exists. (k) t = k = 1. (l) t = 0, no min. (m) Neither exists. (n) t = - 5, no min. 6 (a) Suppose that α and β are least upper bounds for S . Since α = sup S and β is an upper bound for S , α β . Since β = sup S and α is an upper bound for S , β α . Therefore α = β . (b) Same proof. 8 S, T non-empty bounded subsets of R and S T . Choose any s S . Then s T . Therefore s inf T . It follows that inf T is a lower bound for S and therefore inf T inf S . Similarly, s sup T and so sup T is an upper bound for S . Thus, sup S sup T . 10 (a) Between x and y there is a rational number r 1 . Between x and r 1 there is a rational number r 2 . Between x and r 2
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